On the Solvability of Magnetic Differential Equations

The calculation of both resistive and ideal plasma equilibria amounts to solving a number of magnetic differential equations which are of the type $\vec {B}\cdot \nabla \Phi =s$. We apply the necessary and sufficient criterion for the existence of the potential $\Phi$ and find that a static equilibrium configuration of a magnetically confined plasma does not exist in axi-symmetric toroidal geometry.